Assuming all this is correct -- and I am not 100 percent sure it is -- then if I were to apply the same procedure to [itex]e^{ax^2-bx}[/itex] I should end up with the above multiplied by the integral of [itex]e^{-bx}[/itex]. But if I try to integrate that I get infinities.

But I wasn't sure if that was a completely wrong approach, or if I even did the above bit correctly. The thing I have trouble getting my head around is whether e is periodic or not and what period to use when applying Fourier equations. I might be approaching this entirely wrongly.